Numerical treatment of many electron atoms

For pure two-body systems, like the hydrogen atom, it is possible to solve
the Schroedinger equation analytically.
For systems with few electrons, such as helium, the
"many-electron" problem can be solved more or less exactly (at present
the non-relativistic ground state energy of helium is known with fifteen
significant figures). More general many-electron systems cannot be treated with
such precision, however. A majority of the elements in the periodic table are
many-electron systems where the motion of every electron
is coupled to the motion of all the other electrons as well as to the nucleus.
To study such systems we have to rely on some approximation method.

One widely used approximation method is the Hartree-Fock method. It
is based on the rather natural approximation that every electron moves
in the potential created by the nucleus plus the average potential
of all the other electrons. This assumption leads to the
independent-particle model, which essentially reduces the many-electron
problem to the problem of solving a number of coupled single-electron equations.

The single-electron equations are solved in an iterative manner until a chosen
level of self-consistent accuracy is achieved. Hartree made the first
calculation based on these ideas by hand in 1928, but calculations of this type
are, of course, best suited for computers. Today there are several computer
codes available for anyone who is interested in atomic properties. This outline
describes one of the first such codes, written by Herman and Skillman in 1961.

The Hartree-Fock approximation is a fast and reliable method for a wide
range of atomic systems, but it is just a FIRST approximation. There
are several calculation schemes developed to generate improved
results.